Current reservoir simulators are encumbered by the level of detail available for very large, fine-scale reservoir models, which often are composed of millions of grid cells. The quality of reservoir simulation is very dependent on the spatial distribution of reservoir properties; namely porosity and permeability. The permeability of subsurface formations typically displays high variability levels and complex structures of spatial heterogeneity that spans a wide range of length scales. As a result, a large amount of effort towards developing reservoir simulators has been dedicated to better characterizing reservoir properties and developing efficient numerical algorithms for high-resolution models.
Simulation methods employing multi-scale algorithms have shown great promise in efficiently simulating high-resolution models for subsurface reservoirs having highly heterogeneous media. For example, one proposed multi-scale method includes a nested dual grid approach for simulation of both flow and transport in heterogeneous domains. The nested grids are employed to construct a fine-scale flow field and solve the saturation equations along streamlines. Generally, multi-scale methods can be categorized into multi-scale finite-element (MSFE) methods, mixed multi-scale finite-element (MMSFE) methods, and multi-scale finite-volume (MSFV) methods.
Most multi-scale approaches for flow in porous media are designed to develop a coarse-scale pressure equation from the elliptic pressure equation and reconstruct the fine-scale pressure field via basis functions. The hyperbolic (or parabolic) transport equation in fine-scale is then directly or iteratively solved for saturations. The coarsening of the transport equation is much more challenging than that of the elliptic pressure equation. The hyperbolic nature of transport equation entails prolongation and restriction operations of saturation that are strongly dependent on the history of saturation development in a coarse-scale grid with specific underlying heterogeneous permeability distribution. Especially, as the correlation length of permeability is often much larger than the coarse-scale grid size, it is less probable that universal prolongation and restriction operators for saturation can be devised in a functional form of system variables and/or characteristic parameters.
Many approaches have been proposed as alternative computational methods to fine-grid simulation; however, these methods have typically been prone to significant error or have proven to be only slightly less expensive than full simulation of the fine-scale grid. There is a need for a more efficient multi-scale numerical algorithm that can be used to simulate a very large, fine-scale reservoir model. Ideally, the method would provide for accurate interpolation or extrapolation of physical phenomena in one scale to a different scale, such that the effects of the fine-scale are correctly captured on the coarse-scale grid.